臺灣博碩士論文加值系統

本文主要以B-spline函數為基底函數解三維不規則形狀彈性力學問題。從文獻中發現，目前尚無任何研究將B-spline應用於三維不規則形狀彈性力學問題。本文利用二元空間分割法與幾何布林運算判別元素落在實體內或邊界上，若是規則形狀元素則可簡單地以六面體形狀直接積分；針對不規則形狀元素處理的方式，如同有限元素法，將不規則幾何映對至主要元素中，再利用高斯積分法進行積分。
首先對於規則實體分別以兩種不同邊界條件測試，使用B-spline有限元素法可用較少的自由度得到與傳統有限元素法相近的位移、應力分析結果。其後對兩個不規則實體分析，由立方實體內部挖圓柱可知，當元素增多時，除了二階分析結果不甚理想之外，其餘高階基底函數收斂效果都比有限元素法佳；在立方實體內部挖圓球分析結果顯示當網格不夠小時，則顯現不出高階收斂快的效果，且當縮小網格時發現部分元素會產生較複雜的形狀，此部分整理的積分型態種類太多仍待克服。
由實例結果，可以確定了使用B-spline解三維不規則形狀問題分析結果比有限元素法更為準確。

We used the B-spline functions as the basis functions to solve the three dimensional linear elastic problems with irregular shapes. From the references, there are no researchers using B-spline functions to analyze three dimensional linear elastic problems with irregular shapes. Binary space partition method and geometry Boolean operation are used to determine which element is within the domain or across the boundary. If the element is within the domain, then it is regular element with a cube shape and can be integrated easily. If the element is across the boundary, then for the element across boundary, we map the irregular geometry to a master element, a regular cube, as the finite element used. The integration of the irregular shape is integrated in the master element using Gauss quadrature.
First, we analyze a regular cube with two different boundary conditions. Using B-spline finite element method can decrease the degree of freedom and obtain the same displacement and stress analysis results compare to the traditional finite element method. A cube with a cylinder hole and a cube with a sphere hole are used as irregular shapes examples in this thesis. In the former case, we obtained better result accuracy from the high-order B-spline basis function than the results from traditional finite element method except for the second-order B-spline basis function case. In the later case, the problems we analyzed are limited to small elements numbers. In the situation, the effect of high-order B-spline basis function is not shown. We still have problems to integrate these irregular domains. It should be overcome in the future.
From the results in these examples, using B-spline finite element method to solve three dimensional problems with irregular shapes has better accuracy than traditional finite element method.

摘要 I
Abstract II
目錄 IV
誌謝 VI
表目錄 VIII
圖目錄 IX
符號說明 XII
第一章 緒論 1
1.1 前言 1
1.2 研究動機與方法 2
1.3 文獻回顧 3
1.4 論文架構 4
第二章 相關理論 6
2.1 有限元素法於三維彈性力學理論 6
2.2 B-spline曲線介紹 10
2.2.1 非均勻B-spline曲線 10
2.2.2 均勻B-spline曲線 14
2.3 幾何布林運算 15
2.3.1 建立二元空間分割建樹 16
2.3.2 布林運算基本判別式 20
第三章 三維B-spline有限元素法 27
3.1 基底函數 27
3.2 網格化 30
3.3 元素積分方式 32
3.4 施加邊界條件 36
3.5 自由度算法 37
第四章 三維B-spline有限元素法應用範例 38
4.1 規則邊界實體應力分析 39
4.1.1 規則實體一邊界條件 39
4.1.2 規則實體二邊界條件 43
4.2 不規則邊界實體應力分析 47
4.2.1 實體中心挖去圓柱 47
4.2.2 實體中心挖去圓球 60
第五章 結論與建議 64
參考文獻 66
自述 69

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